arXiv subjects

Alina Shalukhina

Publications and source records attributed to Alina Shalukhina.

4 recordsLinked to original sources

arXiv2025

The maximal function on spaces of homogeneous type, or adjacent dyadic cubes do good

We prove that the Hardy--Littlewood maximal operator $M$ is bounded on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$, with $1<p_-\le p_+<\infty$, over an unbounded space of homogeneous type $(X,d,μ)$ with a Borel-semiregular measure $μ$, if and only if the averaging operators $T_\mathcal{Q}$ are bounded on $L^{p(\cdot)}(X,d,μ)$ uniformly over all families $\mathcal{Q}$ of pairwise disjoint ``cubes'' from a Hytönen--Kairema dyadic system on $X$. This extends Diening's well-known characterization of the boundedness of $M$ on $L^{p(\cdot)}(\mathbb{R}^n)$ to the setting of spaces of homogeneous type, while also providing a slight refinement of the original result.

math.CA↗

arXiv2024

Self-improving boundedness of the maximal operator on quasi-Banach lattices over spaces of homogeneous type

We prove the self-improvement property of the Hardy--Littlewood maximal operator on quasi-Banach lattices with the Fatou property in the setting of spaces of homogeneous type. Our result is a generalization of the boundedness criterion obtained in 2010 by Lerner and Ombrosi for maximal operators on quasi-Banach function spaces over Euclidean spaces. The specialty of the proof for spaces of homogeneous type lies in using adjacent grids of Hytönen--Kairema dyadic cubes and studying the maximal operator alongside its dyadic version. Then we apply the obtained result to variable Lebesgue spaces over spaces of homogeneous type.

math.CA↗

arXiv2024

A necessary condition for the boundedness of the maximal operator on $L^{p(\cdot)}$ over reverse doubling spaces of homogeneous type

Let $(X,d,μ)$ be a space of homogeneous type and $p(\cdot):X\to[1,\infty]$ be a variable exponent. We show that if the measure $μ$ is Borel-semiregular and reverse doubling, then the condition ${\rm ess\,inf}_{x\in X}p(x)>1$ is necessary for the boundedness of the Hardy-Littlewood maximal operator $M$ on the variable Lebesgue space $L^{p(\cdot)}(X,d,μ)$.

math.FA↗

arXiv2018

On the extension of the Reverse Hölder Inequality for power functions on the real axis

We consider the class of all non-negative on $\mathbb{R_+}$ functions such that each of them satisfies the Reverse Hölder Inequality uniformly over all intervals with some constant the minimum value of which can be regarded as the corresponding "norm" of a function. We compare this "norm" with the "norm" of an even extension of a function from $\mathbb{R_+}$ on $\mathbb{R}.$ In this paper the upper estimate for the ratio of such "norms" has been obtained. In the particular case of power functions on $\mathbb{R_+}$ the precise value of the increase of the "norm" of its even extension is given. This value is the lower estimate for the analogous one in the case of arbitrary functions. It has been shown that the obtained upper and lower estimates for the general case are asymptotically sharp.

math.CA↗